fluctuations, backtracking and proofreading in transcription

KITP, May 2011

Fluctuations, backtracking and

proofreading in Transcription

Tanniemola B Liverpool

Department of Mathematics

University of Bristol

KITP, May 2011

Acknowledgements

In collaboration with

M Voliotis

(Mathematics, Bristol)

N Cohen and C Molina-París

(Applied Maths & Computing, Leeds)

KITP, May 2011

Plan

1. Fluctuations, pauses and backtracking in

transcription

1. Motivation - stochastic gene expression

2. RNAP: microscopics of transcription

3. Pauses and bactracking single molecule experiments

4. Distribution of elongation times

5. Integrated model of transcription RNA population

dynamics

6. A model for proofreading in transcription

2. Conclusions and outlook

KITP, May 2011

Gene expressionCentral Dogma

DNA RNA proteinTranscription Translation

Gene expression = set of

reactions that control quantities

of gene products (proteins)

Differences in gene expression is

thought to control most

aspects of cellular behaviour

phenotype

However even within related cells of the same phenotype there are

variations in the levels of expressed genes

KITP, May 2011

Variations in gene expressionStandard ‘systems’ viewMicroarray, Northern blot, RT-PCR

experiments

Bulk RNA/protein levels from

‘homogeneous’ population

extracts

Gives impression of gene

expression as continuous

smooth process

BUT highly irregular periods of

activity and inactivity

Important : differences in transcription/translation between related

cells differentiation, disease, …

Fluctuations in space and time

KITP, May 2011

Stochastic Gene expression

Fluctuations quantify variability of

cellular behaviour

Fluctuations can be due to intrinsic

or extrinsic factors (somewhat

vague operational definition) Copy Cat

Twins

New observational techniques

Sources of fluctuations

Macroscopic fluctuations in environment

Variability of internal state of cell

Genetic mutation

Small numbers of macromolecules involved in

gene regulation/expression

Stochastic nature of production and degradation

of RNA transcription products Elowitz et al (2002)

E-Coli

Genetic Savings

and Clone inc.

Texas A&M

Raser & O’Shea (2005)

Modelling FluctuationsChemical Master equations

Discrete reactants , probabilistic chemical reactions

Transcription and translation as one-stage processes

Poisson population statistics

However production/degradation of proteins/mRNA

are multi-stage processes

Transcription 3 main stages

Initiation

Elongation

Termination

How consistent is this with simple exponential

birth/death Markov processes which give Poisson

statistics?

BUT recent expts. tracking RNA

expression levels in single cells see

non-Poissonian fluctuations

Golding et al , Cell (2005)

Alternative processes ?

Raj et al , PLOS Biology (2006)

Zenklusen et al, Nature

Struc. Mol. Biol. (2008)

True for some experiments

Chubb and Liverpool, Current Opinion in Genetics and Dev. (2010)

Chemical master equations

Single timescales

Steady state distribution is Poisson

Prob of n molecules Production rate Degradation rate

COMPLEX

Transcription: DNA mRNA

Greive & von Hippel, (2005)

• Nobel prize for Medicine 1965, F. Jacob,

J. Monod and A. Lwoff (prokaryotic)

• Nobel prize for Chemistry 2006, Roger

Kornberg (eukaryotic RNAP)

Termination

RNA polymerase (~ 150 KDa)

Initiation

Elongation

Benoit Coulombe (Montreal)

Optical tweezer experimentsPauses of E-Coli RNAP observed in-vitro

Operationally experiments classified into

‘short’ (< 20s) and ‘long’ pauses

Backtracking of E-Coli RNAP observed

during long pauses.

Shaevitz et al, Nature, 426, 684 (2003)

RNAP transcribes with

remarkably low error rate in-vivo

Has recently been suggested that

back-tracking is involved in

proofreading

membraneSingle molecule experiments

Backtracking rearwards motion of

RNAP along DNA template in direction

opposite to normal elongation

Shift of transcription bubble, however

DNA-RNA hybrid remains in register, 3’

end of RNA moves away from active site

Broad non-exponential temporal

distribution of ‘long’ pauses

Elongation phase

membrane Pipe dreams …

Dynamical question - To understand how the complex

behaviour of cells controlled by the expression of their

genes emerges from their components.

????????????????

Some much smaller goals - using simple physical models …

•Can we understand something about the origin of intrinsic

fluctuations from the bottom up ?

• Can we link single molecule behaviour to gene expression

experiments at the cellular level ?

KITP, May 2011

A theoretical model of transcription

Initiation - Poisson process

Elongation phase

• Position of last transcribed nucleotide

n (size of transcribed mRNA)

• Position of polymerase active site

relative to n m

-n < m < 1

m=0 pre-translocated

m=1 post-translocated

m<0 backtracked

• Polymerisation ( + new nucleotide) only from post-translocated state

• Depolymerisation ( - new nucleotide) only from pre-translocated state

• Initiation (n=0) and termination (n=N)

Based on biochemical mechanism proposed by Yager and von Hippel, Biochemistry, 30, 1097 (1991)

KITP, May 2011

Model of transcription elongation

•Simulations got similar patternsVoliotis et al, Biophys. J, 94, 334 (2008)

First passage problem

Schematic of state transitions

Backtracking restricted to -M >-n (hairpins,cleavage,…)

We want to find the statistics of the elongation time

(time to get from n=0 to n=N)

KITP, May 2011

Model A: no backtracking

•Simulations got similar patternsVoliotis et al, Biophys. J, 94, 334 (2008)

We want to find the statistics of the

elongation time (time to get

from n=0 to n=N)

Schematic of state transitions

Backtracking restricted to -M >-n

(hairpins,cleavage,…)

Mean field approximation:

biased random walk

Reflecting BC (n=0)

Absorbing BC (n=N)

First passage problem easy

Model A :translocation limited polymerisation

KITP, May 2011

Model A: no backtracking

• Under normal conditions, polymerisation overwhelmingly favoured over

depolymerisation

• Mean elongation time, t and variance t

M Voliotis et al, Biophys. J, 94, 334 (2008)

We want to find the statistics of the elongation time (time to get from n=0 to n=N)

• Compare with Monte Carlo simulations of full model

• For large N, approaches a Gaussian

• Fluctuations smaller than exponential process ( mRNA Population sub-Poisson)

KITP, May 2011

A model for backtracking pauses Before we include pauses in our dynamics we need a model for

backtracking pauses themselves.

Can we explain the broad distribution of pause durations?

• A pause starts when the TEC enters the state m=-1 from the state m=0

• From m=-1, TEC hops across backtracked states with hopping rate c

• Because of hairpins, RNA-DNA interactions backtracking is restricted

and proceeds up to m=-M for M < n and up to m=-n for n > M

KITP, May 2011

Dynamics of backtracking pausesUse new variable l=-m where 1 ≤ l ≤M

Probability of finding polymerase starting at l=1 at t=0 at position l at time

t is P(l,t)

• Obtain the time to terminate a pause by returning to state l=0

• First passage problem in a finite domain with reflecting BC’s

• Probability flux to the state l=0 probability of exiting the pause at time t

• Can do a similar calculation when there is transcript arrest (absorbing BC’sboth sides)

(reflecting)

(absorbing)

KITP, May 2011

Dynamics of backtrackingSeries & asymptotic expansion of 1 function used to obtain limiting behaviour.

Mean pause duration

Power law behaviour for t << M2/c

heavy-tailed distribution

observed by Shaevitz et al.

M Voliotis et al, Biophys. J, 94, 334 (2008)

KITP, May 2011

Single molecule experiments

Power law with t-3/2 ?

Recent quantitative experiments of pause distributions of

eukaryotic RNAP II Galburt et al, Nature (2007)

-3/2

KITP, May 2011

Model B: transcription with pausesNow we are in the position to study a

model for elongation with

pauses

Pauses dominate elongation time

Pauses negligible - model A

Macroscopic observables

o Number of pauses over a

DNA template of length N

o Sum of lifetime of pauses

relative to time spent on active

polymerisation

KITP, May 2011

Model B: transcription with pauses

As R , distribution becomes broader

m

When translocation not the rate-limiting step

Pauses negligible

If polymerisation favoured

Pauses dominate

Define

KITP, May 2011

Integrated model of mRNA productionWe are really interested in mRNA levels in the cell Need to consider

production (transcription) and degradation

Initiation + model B + termination(fast) + degradation

Polymerases cannot get too close because of

additional work to deform the DNA helix

‘Christmas Tree’

S.L. Gotta et al, J. Bacteriol., 173, 6647 (1991)

Many initiations can occur in the time to produce

a single RNA multiple occupation of DNA

template by TECs moving in tandem

Each polymerase synthesizing a nascent mRNA

KITP, May 2011

Model of mRNA production

Active site of a TEC with

(n,m) is at position

Minimum (exclusion)

distance between TEC’s

Relevant timescales

Time for transcription initiation

Time needed by the TEC to transcribe L nucleotides

The mean time of a backtracking pause

Study numerically using MC simulations

KITP, May 2011

Model of mRNA production

(Super-Poisson)

When initiation is the rate limiting step

(approx Poisson)

Long pauses dominate transcription

Polymerisation is the rate limiting step

(Sub-Poisson )

Bursts of RNA production due to rare and long-

lived pauses of TECs acting as congestion

points

Switches between states of high and low

mRNA production

KITP, May 2011

Relation to ASEP?

1-d non-equilibrium model been subject of much study

Two species model + particles

Symmetry breaking transition

The Asymmetric Simple Exclusion Process

- particles

Switching between left and right moving states

B. Derrida, Phys. Rep., 301, 65 (1998).

M.R. Evans et al, PRL, 74, 208 (1995)

symmetric

asymmetric

KITP, May 2011

mRNA populations in E-Coli

Direct measurement of mRNA populations

MS2-GFP fusion protein

Golding et al, Cell, 123, 1025 (2005)

mRNAs with 96 MS2 binding site

Numbers

Rate of entering back-tracking state

= rate of NTP misincorporation: 1 error/kbp

Shaevitz et al (2005)

mRNA degradation rate: 0.014/min

Depolymerisation rate : two orders magnitude lower

Golding et al (2005)

Polymerisation rate: 25bp/s-50 bp/s

Initiation rate ≈ 0.1/s bursting .

Rate of backtracking diffusion c: 0.1-0.2 bp/s

But what about stochastic dynamics of transcription factor binding ?

KITP, May 2011

Backtracking & error correction …

Hopfield, Proc. Natl. Acad. Sci. (1974)

Ninio, Biochimie 57, 587 (1975)

Thermodynamic fraction of misincorporated nucleotides

Observed transcriptional error fraction

Kinetic proofreading : there must exist

an active mechanism of error correction

Requires a branching

process

Possible mechanism -

circumstantial evidence

…

Backtracking

mRNA cleavage (Gre,

TFIIS)

Limiting error fraction

KITP, May 2011

Backtracking & error correction …

Hopfield, Proc. Natl. Acad. Sci. (1974)

Ninio, Biochimie 57, 587 (1975)

Thermodynamic fraction of misincorporated nucleotides

Observed transcriptional error fraction

Kinetic proofreading : there must exist

an active mechanism of error correction

Requires a branching

process

Possible mechanism -

circumstantial evidence

…

Backtracking

mRNA cleavage (Gre,

TFIIS)

KITP, May 2011

Backtracking & error correction

Voliotis et al, PRL, 102, 258101 (2009)

Polymerise and

fix error

Cleave and

correct error

Renormalized Error fraction : ratio

of wrong to correct nucleotides

KITP, May 2011

A model of error correction


State of TEC given by (n,m)

- last transcribed nucleotide

- position of active site relative to n

Error at position

At each nucleotide position (n,0) :

correct nucleotide

wrong nucleotide

backtrack

Or ?

Cleave at rate

KITP, May 2011



Quantities of interest that characterise

the different competing processes

Many possible energy landscapes, take

for example …

KITP, May 2011


M+2 possible outcomes

Polymerise correct nucleotide (n,0) → (n+1c,0)

Polymerise wrong nucleotide (n,0) → (n+1w,0)

Cleave from any backtracked state (n,m=l>0) → (n-l,0)

Using Laplace transform techniques we can obtain the splitting probabilities pi

for each of the m+2 outcomes as well as their conditional mean exit time τi

Dynamics described using Master eqn.

Where s is a binary string that keeps track of correct and wrong nucleotides along the nascent mRNA,

si=0 - wrong nucleotide at position i

si=1 - correct nucleotide at position i

W(s) - denotes the dependence of the rates on the sequence of correct and wrong nucleotides.

KITP, May 2011


Unrealistic to think that backtracking rate unchanged by error and

Presence of error leads to ‘renormalization’ of K

Error fractionFor

Can have effective when no error present

boundary M M attempts to correct an error

KITP, May 2011

Numbers

Galburt et al, Nature (2007); Shaevitz et al, Nature (2003)

Relative backtracking rate, K ~ 0.1

Spontaneous error fraction : 10−2 − 10−3 (∆G ≈ 4 − 7kB T ) [1].

The cleavage rate : 0.1 − 1s−1 for bacterial RNAP in the presence of saturating concentrations of accessory cleavage factors

Hopping rate : 1 − 10 s−1 [3,4].

Relative backtracking rate, K ~ 0.1

Blank et al, Biochemistry (1986)

Sosunova et al, PNAS (2003)

KITP, May 2011

Conclusions

– single step birth/death models not sufficient to model

fluctuations in gene transcription

– a model of back-tracking pauses show that they can play a

significant role in determining fluctuations

– Single step models valid if initiation is rate-limiting step

– Inclusion of pausing dynamics with multiple RNAs on

DNA template leads to bursting dynamics

– Backtracking can be used as model for proofreading in

transcription

KITP, May 2011

Perspectives

elongation under force …

transcript arrest in integrated model …

sequence dependence …

translation ..

bursting dynamics in gene expression …

fluctuations, backtracking and proofreading in transcription

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